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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p><dfn class="terminology">Solution:</dfn> Consider</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq4_18.html ./knowl/eq4_17.html">
\begin{equation}
y^{\prime \prime \prime}-y^{\prime}=0.\tag{4.4.12}
\end{equation}
</div>
<p class="continuation">The characteristic equation is</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq4_18.html ./knowl/eq4_17.html">
\begin{equation*}
r^3-r=0 \to r(r+1)(r-1)=0 \to r_1=0, ~r_2=1, ~r_3=-1.
\end{equation*}
</div>
<p class="continuation">The general solution to (<a href="" class="xref" data-knowl="./knowl/eq4_18.html" title="Equation 4.4.12">(4.4.12)</a>) is</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq4_18.html ./knowl/eq4_17.html">
\begin{equation*}
y=C_1+C_2 e^x+C_3 e^{-x}.
\end{equation*}
</div>
<p class="continuation">To find a particular solution to (<a href="" class="xref" data-knowl="./knowl/eq4_17.html" title="Equation 4.4.11">(4.4.11)</a>), we use the formula we have derived</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq4_18.html ./knowl/eq4_17.html">
\begin{equation*}
Y=\sum_{m=1}^3 y_m \int \frac{g(x) W_m}{W(y_1, y_2, y_3)} \textrm{d} x.
\end{equation*}
</div>
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